Spontaneous wave function collapse models (CM) have been proposed 30 years ago as a possible solution of the quantum measurement problem, i.e. the apparent contradiction between the unitary evolution and the reduction/collapse postulate. CM are based on a stochastic modification of the Schrodinger equation, which naturally leads to the localization (spatial collapse) of macroscopic objects and to the emergence of definite outcomes and the Born rule in any measurement process. In contrast to philosophical interpretations, collapse models can be experimentally tested, as they assume that quantum theory, similarly to newtonian gravity, is not an exact theory. Specifically, two main experimentally testable effects emerge. The first is a fundamental breakdown of the superposition principle, which can be tested by observing larger and larger superpositions in matter-wave interferometry. The second is a tiny violation of energy conservation, leading to a universal heating or force noise acting on any mechanical system. The latter approach is currently setting the strongest constraints in the CM parameter space. In this talk I will discuss how ultralow force noise mechanical experiments can be used to derive upper bounds on collapse models parameters, with special emphasis on the most general model, the Continuous Spontaneous Localization (CSL). I will first discuss the current bounds that can be inferred by recent experiments with ultracold kHz frequency cantilevers and by the space mission LISA Pathfinder, and then outline future prospects. One of the most ambitious attempts to improve the bounds on collapse models is being pursued in Southampton, and is based on monitoring the diffusion of levitated nanoparticles at very low temperature.